When entries in /etc/cron.weekly are run, what user/group are they
run as?

Whatever user it says in /etc/crontab. On mine it is root.

Odd. I'm getting an error running popularity-contest because it says it
doesn't have permission to run dpkg. According to /etc/crontab, it's
running as root, and root certainly does have permission to run dpkg!

Additionally, popularity-contest only uses dpkg for "dpkg-query --show
--showformat='\${status} \${package}\\n'|" which should even work as a
normal user. Indeed, the only reason it[pop-con] appears to need to run
as root is to write to /var/log/popularity-contest*[why not
/var/(lib|cache)/popularity-contest/?], but I could be wrong.